Index

We have already discovered the concepts of collineations, automorphisms and isomorphisms.

Recall that we have defined an isomorphism between two geometries as a 1-1 mapping from one geometry to another such that the point set of one geometry is mapped to the point set of another which induces a 1-1 correspondance between the line sets of each geometry and the incidence structure is preserved. In addition, we defined the isomorphism of a geometry onto itself as an automorphism. Many specialists in the field of finite geometry refer to automorphisms of a finite geometry as collineations, especially when the geometries are linear spaces: collineations have many interesting and useful properties.

In Problem 6 we showed that there are 168 collineations of the Fano plane. We have also seen that the set of collineations on a near-linear space (and any projective plane is a linear space) is a group. Thus, we can apply some of the terms and theorems that we know about groups to the set of collineations:

Recall your earlier reading assignments from Berger & Dorwart, as well as Yaglom, where perspectivities on the real projective plane were discussed.

The identity mapping will always be a central collineation, as the identity fixes EVERY point of the plane. In fact, the center is never the only point that is fixed in a central collineation, but unless the collineation is the identity map, there will be only one center point (can you prove this?).

We'll now present an example of how exciting the duality principle is. We know that the duality principle holds for any given projective plane, meaning that given a true statement about a projective plane, we can freely interchange the words "point" and "line" and still have a true statement. So the dual defintion of central collineation would be:

So is the following a true statement? "A collineation f on a projective plane is a central collineation if and only if it is an axial collineation." Explain.

Does it automatically follow from what we said above about centers, that unless the axial collineation is the identity map, that there can only be one axis? Explain.

Can you always play this "dual defintion" game when working with projective planes?

Note, however, that the center does not necessarily lie on the axis (can you come up with an example?). When the center lies on the axis, we call the perspectivity an elation. When the center is not on the axis, we call the perspectivity a homology.

Given a perspectivity on a plane with unique center c and axis l, we will call this a (c,l) -collineation. So is any central or axial collineation that is not the identity mape a (c,l)-collineation?

We will see below that giving the point c and the line l does not uniquely determine a persepctivity.

Recall from our math history readings on the real projecitve plane, that mathematicians could compose perspectivities together to get another perspectivity. You might find it enlightening to show that, under the operation of composition, the set of (c,l)-collineations of a given projective plane does form a group.

We now propose that any collineation can be uniquely determined provided that we are given a projective plane, the center c, the axis l, and its effect on one other point p that is distinct from c and that does not lie on l.

This is easy to see, as if p does not lie on l, then f(p) also does not lie on l (by definition of axis). Let q be another point such that q does not lie on the same line as c and p. Let the intersection of l and the line pq be another point r (thus r lies on l and therefore remains fixed throughout the collineation). We then know that f(q) is the intersection of the line cq with the line f(p)r. This must be true because we know that p and q and r all lie along a line in the projective plane. Thus f(p), f(q) and f(r) = r must also all lie along the same line. The line cq is simply the same line that is incident with f(q). Thus, we can show how a central collineation will act on any point that does not lie on the line cp.

If however the point q is on the line cp, then simply choose a point p' that is not on the line cp. Then q is not on this line, and the same argument as above holds.

Use the Fano plane, the projective plane of order 2, to demonstrate this construction for yourself.